text: We will present a simplified version of the calculation. It differs from the full calculation in three ways:
  # The ideal gas consists of particles confined to one spatial dimension.  
  # We keep only the terms of order , dropping all terms of size n or less, where n is the number of particles. For our purposes, this is enough, because this is where the Gibbs paradox shows up and where it must be resolved. The neglected terms play a role when the number of particles is not very large, such as in computer simulation and nanotechnology. Also, they are needed in deriving the Sackur–Tetrode equation.    
  # The subdivision of phase space into units of the Planck constant (h) is omitted. Instead, the entropy is defined using an integral over the "accessible" portion of phase space. This serves to highlight the purely classical nature of the calculation.
  We begin with a version of Boltzmann's entropy in which the integrand is all of accessible phase space:
  The integral is restricted to a contour of available regions of phase space, subject to conservation of energy.  In contrast to the one-dimensional line integrals encountered in elementary physics, the contour of constant energy possesses a vast number of dimensions. The justification for integrating over phase space using the canonical measure involves the assumption of equal probability.  The assumption can be made by invoking the ergodic hypothesis as well as the Liouville's theorem of Hamiltonian systems.
  (The ergodic hypothesis underlies the ability of a physical system to reach thermal equilibrium, but this may not always hold for computer simulations (see the Fermi–Pasta–Ulam–Tsingou problem) or in certain real-world systems such as non-thermal plasmas.)
  Liouvilles theorem assumes a fixed number of dimensions that the system explores'.  In calculations of entropy, the number dimensions is proportional to the number of particles in the system, which forces phase space to abruptly change dimensionality when particles are added or subtracted. This may explain the difficulties in constructing a clear and simple derivation for the dependence of entropy on the number of particles.
  For the ideal gas, the accessible phase space is an (n − 1)-sphere (also called a hypersphere) in the n-dimensional  space:
  To recover the paradoxical result that entropy is not extensive, we integrate over phase space for a gas of  monatomic particles confined to a single spatial dimension by . Since our only purpose is to illuminate a paradox, we simplify notation by taking the particles mass and the Boltzmann constant equal to unity: . We represent points in phase-space and its x and v parts by n and 2n' dimensional vectors:
  :    where       and   
  To calculate entropy, we use the fact that the (n-1)-sphere,  has an -dimensional "hypersurface volume" of
  For example, if n = 2, the 1-sphere is the circle , a "hypersurface" in the plane. When the sphere is even-dimensional (n odd), it will be necessary to use the gamma function to give meaning to the factorial; see below.